Mathematics and Statistics Vol. 14(2), pp. 189 - 205
DOI: 10.13189/ms.2026.140201
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New Results Supporting the Validity of the Lindelöf Hypothesis Using the Maynard–Guth Hamiltonian

Zeraoulia Rafik ^1,2,*, Simeon Casanova Trujillo ³, Alvaro Humberto Salas ^3
1 Djilali Bounaama University of Khemis Miliana, Algeria
² Laboratory of Pure and Applied Mathematics (C1151600), University of Laghouat, Algeria
³ Department of Mathematics, Universidad Nacional de Colombia, Bogotá, Colombia

ABSTRACT

This paper develops a quantum–spectral model for prime counting in almost-short intervals, motivated by the Maynard–Guth estimate and the long-standing Riemann and Lindelöf hypotheses. We refine a perturbed Schrödinger-type Hamiltonian whose potential encodes both the main term and the exponentially decaying error term of the prime number theorem in these intervals, and we analyze its uncertainty relation, symmetry/self-adjoint structure, and spectral behavior. Numerically, we compute eigenvalues and eigenfunctions using finite-difference and finite-element discretizations on large domains and compare the unfolded spacing statistics with random-matrix predictions; we further probe Lindelöf-type boundedness by evaluating at arguments derived from the computed spectrum. The resulting spectrum is real, exhibits symmetry and band structure, and shows level repulsion compatible with a Hilbert–Pólya operator candidate, while the zeta evaluations remain bounded (and frequently close to 1) in the tested ranges. The main contribution is an explicit bridge from Maynard–Guth prime-density/error terms to a concrete Hamiltonian framework that yields testable spectral and thermodynamic predictions. Limitations include finite discretization ranges and the use of conjectural inputs in parts of the prime-gap discussion; no claim of a proof is made, but the results provide quantitative guidance for future rigorous and computational work.

KEYWORDS
Prime Numbers, Almost-short Intervals, Maynard–Guth Theorem, Quantum Hamiltonian, Riemann Zeta Function, Lindelöf Hypothesis, Hilbert–Pólya Conjecture

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Zeraoulia Rafik , Simeon Casanova Trujillo , Alvaro Humberto Salas , "New Results Supporting the Validity of the Lindelöf Hypothesis Using the Maynard–Guth Hamiltonian," Mathematics and Statistics, Vol. 14, No. 2, pp. 189 - 205, 2026. DOI: 10.13189/ms.2026.140201.

(b). APA Format:
Zeraoulia Rafik , Simeon Casanova Trujillo , Alvaro Humberto Salas (2026). New Results Supporting the Validity of the Lindelöf Hypothesis Using the Maynard–Guth Hamiltonian. Mathematics and Statistics, 14(2), 189 - 205. DOI: 10.13189/ms.2026.140201.